During an appearance on CNN's "State of the Union" on Sunday, Sen. Jim DeMint (R-S.C.) said any "presidential candidate who's not willing to say we have to balance our budget should not be president of the United States."

The South Carolina senator -- a conservative force who waded into a handful of high-profile GOP primary races in the last election cycle -- recently made clear it'd be a "dealbreaker" for earning his support if a presidential contender doesn't stand behind the "Cut, Cap and Balance" pledge, which entails opposing any debt limit increase without significant spending cuts, enforceable spending caps and congressional approval of a balanced budget amendment.

Republican White House hopefuls former Minnesota governor Tim Pawlenty, U.S. Rep. Ron Paul (R-Texas), former Godfather's Pizza CEO Herman Cain and former U.S. Senator Rick Santorum have all added their names to the pledge.

Former Utah governor Jon Huntsman recently signaled he will not join the ranks of his Republican rivals, saying, "Other than the pledge of allegiance, I don't do a whole lot of pledges."

“Gov. Huntsman believes his own record and the record of all candidates are stronger than any pledge, and his record is one of passing the biggest tax cut in Utah history, implementing a flat tax and balancing the budget,” Sawyer said.

In weighing in on the field of GOP presidential contenders, DeMint said he likes "a lot" of the names in the primary mix and that many of them are reaching out to him.

"I'm working with people in Iowa and New Hampshire and South Carolina, to try to get a lot of people to hold back and not commit so we can see how they respond to this debt ceiling, the balanced budget and some of the things that we're going to face here over the next few months," he explained. "I think we'll know who our candidate is by how they lead based on what we're doing here."